116 BOOK OF MATHEMATICAL PROBLEMS, 



the chord through a, ft is 



X a+ft y . a + ft a-ft 



- cos + ~ sm . = cos - 

 a '2 b 2 '2 



and the intersection of tangents at a, /?, 



"V V 



The polar of (X, Y) is - 8 + ~- = 1 ; and the equation of the two 



8 

 tangents through (JT, Y) is 



It follows from the equation of the tangent that, if the equa- 

 tion of any straight line be Ix + my = 1, and I, m satisfy the 

 equation a a l* + b*m* = 1, the straight line touches the ellipse 



X s V* 



+ y _-s = 1, a result often useful. 



a o 



The equation of the tangent in the form 



x cos 6 + y sin = ^(a 3 cos 2 B + b 3 sin 2 6) 

 may be occasionally employed with advantage. 



The points a, ft are extremities of conjugate diameters if 



a~/2 = ^. Any two points are called conjugate, if each lies on 



2 



the polar of the other ; and any two straight lines, if each passes 

 through the pole of the other. 



If (JT, 7) be the pole of the chord through a, ft it will be 

 found immediately that 



sin a sin j3 cos a cos $ _ sin a + sin ft cos a + cos ft _ 



2Y ~ %X ~ X* Y 2 ' 



" ~^T ~tf + ~ 



